The meaning of "amendment" here does indeed correspond with its generic one of "formal change to legal text". Generically a motion is the proposal under debate. Unlike a bill, which might be tens or even hundreds of pages long, a motion will often be relatively short and simple. Picking a few examples from last week's Hansard:
They might also attempt to progress legislation:
For motions which the standing Commons rules deem amendable, MPs other than the one making the original proposal may propose changes to the language of the motion, ranging from adding or removing a single word, to rewriting the content entirely. The Speaker will then choose which amendments are debated along with the main text. Votes on amendments happen in sequence (again chosen by the Speaker) with a final vote on the resulting motion.
In regard to the EU Withdrawal Act, the enacted version required that if the House decided to reject the government's proposed withdrawal agreement then
- A Minister of the Crown must, within the period of 21 days beginning with the day on which the House of Commons decides not to pass the resolution, make a statement setting out how Her Majesty’s Government proposes to proceed in relation to negotiations for the United Kingdom’s withdrawal from the EU under Article 50(2) of the Treaty on European Union.
- A statement under subsection (4) must be made in writing and be published in such manner as the Minister making it considers appropriate.
- A Minister of the Crown must make arrangements for—
- a motion in neutral terms, to the effect that the House of Commons has considered the matter of the statement mentioned in subsection (4), to be moved in that House by a Minister of the Crown within the period of seven Commons sitting days beginning with the day on which the statement is made, and
- a motion for the House of Lords to take note of the statement to be moved in that House by a Minister of the Crown within the period of seven Lords sitting days beginning with the day on which the statement is made.
It's this "amendable neutral motion" into which the the Labour front bench and various groups of backbenchers, including Yvette Cooper and Nick Boles are attempting to construct instructions for parliament and the government regarding how to proceed with Brexit. Any version with a clear majority probably wouldn't have legal force in of itself, but could modify the rules the Commons operate under and would be dangerous for the government to ignore.
Neither the original text of the government motion, nor the amendments seem to be on the web yet (and there is still time for both to be revised), but from media reports the amendment tabled by Cooper and Boles contains provisions to eventually force a vote by MPs to request an extension on Brexit day if an agreement has not been voted through by February 26. It would do this by allowing time for debate of a bill Nick Boles has already proposed (or a very similar one). This bill would itself have legal force.